Question

There are two containers of the same volume,first container half-filled with sugar syrup and the second container half-filled with milk.Half the content of the first container is transferred to the second container,and then the half of this mixture is transferred back to the first container.Next, half the content of the first container is transferred back to the second container.Then the ratio of sugar syrup and milk in the second container is

• A. \(5\ratio6\)
• B. \(5\ratio4\)
• C. \(6\ratio5\)
• D. \(4\ratio5\)

Answer 1

The Correct Option is A

Let the volume of each container be V.

Initially, both containers are half-filled:

• First container: \( \frac{V}{2} \) sugar syrup
• Second container: \( \frac{V}{2} \) milk

Step 1: Transfer half the content from the first to the second container

• Sugar syrup transferred: \( \frac{V}{4} \)
• Now, second container has:
  • Sugar syrup: \( \frac{V}{4} \)
  • Milk: \( \frac{V}{2} \)
• Total in second container: \( \frac{3V}{4} \)

Step 2: Transfer half of this mixture back to the first container

• Mixture transferred: \( \frac{3V}{8} \)
• Proportions in the mixture:
  • Sugar syrup part: \( \frac{V}{4} \)
  • Milk part: \( \frac{V}{2} \)
• Sugar syrup transferred back: \( \frac{V}{4} \times \frac{1}{2} = \frac{V}{8} \)
• Milk transferred back: \( \frac{V}{2} \times \frac{1}{2} = \frac{V}{4} \)

Step 3: Transfer half of the first container back to the second

• Now in first container:
  • Sugar syrup = remaining = \( \frac{V}{2} - \frac{V}{4} + \frac{V}{8} = \frac{5V}{8} \)
  • Milk = \( \frac{V}{4} \)
• Total in first container = \( \frac{5V}{8} + \frac{V}{4} = \frac{7V}{8} \)
• Half transferred = \( \frac{7V}{16} \)
• Sugar syrup transferred = \( \frac{5V}{8} \times \frac{1}{2} = \frac{5V}{16} \)
• Milk transferred = \( \frac{V}{4} \times \frac{1}{2} = \frac{V}{8} \)

Final content in the second container:

• Existing sugar syrup = \( \frac{V}{4} \)
• New sugar syrup added = \( \frac{5V}{16} \)
• Total sugar syrup = \( \frac{9V}{16} \)
• Remaining milk in second container = \( \frac{V}{2} - \frac{V}{4} = \frac{V}{4} \)
• New milk added = \( \frac{V}{8} \)
• Total milk = \( \frac{5V}{16} \)

Final Ratio:

\[\text{Sugar Syrup : Milk} = \frac{9V}{16} : \frac{5V}{16} = \frac{9}{5}\]

Final Answer: Ratio = \( \mathbf{9:5} \)